Uniform convexity of unitary ideals

ItE is a symmetric Banach sequence which isq-concave with the constant equal to 1 (where 2≦q<∞), thenS _E isq-PL-convex. IfE isq-concave andp-convex with the constants equal to 1 (where 1<p≦2≦q<∞), thenS _E is uniformly convex with modulus of convexity of power typeq and uniformly smooth with modulus of smoothness of power typep.     

Spectrum of the Cesàro operator in ℓ p

We present a novel proof of the fact that the spectrum of the Cesàro operator acting in ? ^p , for 1 < p < ∞, consists of the closed disc centered at q/2 and with radius q/2, where q is the conjugate index of p.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

On linear codes whose weights and length have a common divisor

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1<r<q=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r<q and whose dual minimum distance is at least 3, has length at least (r−1)q+(p−1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.     

Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

New regularity theorems for non-autonomous variational integrals with (p, q)-growth

One prominent problem in the calculus of variations is minimizing anisotropic integrals with a (p, q)-elliptic density F depending on the gradient of a function ${w : \Omega \rightarrow \mathbb{R}^N}$ with ${\Omega \subset \mathbb{R}^n}$ . The best known sufficient bound for regularity of solutions is q <?p (n?+?2)/n. On the other hand, if we allow an additional x-dependence of the density we have the much weaker result q <?p (n + 1)/n. If one additionally imposes the local boundedness of the minimizer, then these bounds can be improved to q <?p?+?2 and q <?p?+?1. In this paper we give natural assumptions for F closing the gap between the autonomous and non-autonomous situation.     

Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lp,s,γ to Lq,r,γ, 1<p<q<∞, 1?r?s?∞, and from L1,r,γ to Lq,∞,γ≡WLq,γ, 1<q<∞, 1?r?∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.     

A Transference Result of the L p Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms

In this paper we develop a transference method to obtain the Gaussian-Riesz transform’s L ^p -continuity and the Laguerre-Riesz transform’s L ^p -continuity from the L ^p -continuity of the Jacobi-Riesz transform, in dimension one, using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials.     

The maximum number of q-cliques in a graph with no p-clique

Let $\text{?(n, p, q)}$ be the maximum possible number of q-cliques among all graphs on n nodes with no p-clique. Turán, in 1941, determined $\text{?(n, p, 2)}$ for all n and p. For each n and p, he found the unique graph which attains this maximum. In this paper we determine $\text{?(n, p, q)}$ for all values of n, p and q. We show that, except for the trivial case 1 ? n < q, Turán's graph is the unique graph which attains the maximum $\text{?(n, p, q)}$ for all q such that 1 < q < p.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

On boundedness of discrete multilinear singular integral operators

Let m(ξ,η) be a measurable locally bounded function defined in R². Let 1?p₁,q₁,p₂,q₂<∞ such that pi=1 implies qi=∞. Let also 0<p₃,q₃<∞ and 1/p=1/p₁+1/p₂−1/p₃. We prove the following transference result: the operator

     

A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

We study the solvability of a semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces H_l,p,q, 1<p<∞, depending on a complex parameter q. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.